Journal of Modern Applied Statistical Methods Volume 7 | Issue 1
Article 2
5-1-2008
Confidence Intervals for the Squared Multiple Semipartial Correlation Coefficient James Algina University of Florida,
[email protected]
H. J. Keselman University of Manitoba,
[email protected]
Randall D. Penfield University of Miami,
[email protected]
Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Algina, James; Keselman, H. J.; and Penfield, Randall D. (2008) "Confidence Intervals for the Squared Multiple Semipartial Correlation Coefficient," Journal of Modern Applied Statistical Methods: Vol. 7 : Iss. 1 , Article 2. DOI: 10.22237/jmasm/1209614460 Available at: http://digitalcommons.wayne.edu/jmasm/vol7/iss1/2
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Journal of Modern Applied Statistical Methods May, 2008, Vol. 7, No. 1, 2-10
INVITED ARTICLES Confidence Intervals for the Squared Multiple Semipartial Correlation Coefficient
James Algina
H. J. Keselman
University of Florida
University of Manitoba
Randall D. Penfield University of Miami
The squared multiple semipartial correlation coefficient is the increase in the squared multiple correlation coefficient that occurs when two or more predictors are added to a multiple regression model. Coverage probability was investigated for two variations of each of three methods for setting confidence intervals for the population squared multiple semipartial correlation coefficient. Results indicated that the procedure that provides coverage probability in the [.925, .975] interval for a 95% confidence interval depends primarily on the number of added predictors. Guidelines for selecting a procedure are presented. Key words: Squared multiple semipartial correlation; effect size; asymptotic and bootstrap confidence intervals. Introduction one independent variable
A commonly used effect size (ES) in multiple regression analysis is the increase in R 2 when
(X ) j
is added to the
model. This ES, which is called the squared semipartial correlation coefficient, and is often symbolized by ΔR 2 , measures the strength of relationship between X j and the dependent
James Algina (
[email protected]) is a Professor of Research and Evaluation Methodology. His research interests are in applied statistics and psychometrics. Email him at
[email protected]. H. J. Keselman is a Professor of Psychology. His research interests are in applied statistics. Email him at
[email protected]. Randall D. Penfield is an Associate Professor of Education. His research interests are in educational measurement and psychometrics. Email him at
[email protected].
variable (Y ) , controlling for the other independent variables in the model. This coefficient ΔR 2 can also be used when several variables are added to the model. In this context, ΔR 2 is called the squared multiple semipartial correlation coefficient (Pedhazur, 1997) or the semipartial R 2 (Cohen, Cohen, West, & Aiken, 2003) and measures the strength of association between Y and the added independent
2
ALGINA, KESELMAN, & PENFIELD variables, controlling for the other independent variables in the model. Hedges and Olkin (1981) presented methods for calculating the asymptotic sampling covariance matrix for commonality components (See also Mood, 1969, 1971). These results can be used to construct a confidence interval (CI) for Δρ 2 , the population ES estimated by ΔR 2 . Olkin and Finn (1995) presented a method equivalent to the Hedges and Olkin method, and illustrated the new method for the case in which there is one independent variable in the model in addition to X j (i.e., for the case of two
investigate, the number of predictors in the full model ranged from p = 3 to p = 9 in steps of 2. After reviewing the results we added conditions with p = 8 predictors in the full model. The difference in the number of predictors in the full and reduced models ( i.e., p − k ) ranged from 2 to p − 1 in steps of 1. The squared multiple correlation coefficient for the predictors in the reduced model ranged from .00 to .50 in steps of .10. The squared multiple correlation coefficient for the full model ranged from ρ r2 to ρ r2 + .10 in steps of .01 and from ρ r2 + .10 to ρ r2 + .20 in steps of .05. Sample size ranged from 50 to 200 in steps of 50. The predictors and the dependent variable were distributed as a multivariate normal distribution. Each of the 7176 combinations of p, k, 2 n, ρr , and ρ 2f was replicated 5000 times in
independent variables). Alf and Graf (1999) simplified the method and showed how to apply it in the general case of p predictors, and Graf and Alf (1999) developed a computer program that computes the CI. Algina and Moulder (2001) found that when the squared semipartial correlation coefficient is of interest, researchers would need very large samples sizes (n) to achieve adequate coverage probability for Δρ 2 . Algina, Keselman, and Penfield (2007) found that it was possible to obtain much better coverage probability, with smaller sample sizes, if percentile bootstrapping methods were used for setting CIs for the squared semipartial correlation coefficient, rather than relying on the asymptotic intervals. The purpose of the present paper was to investigate whether asymptotic or percentile bootstrap intervals would result in adequate coverage probability for Δρ 2 when a squared multiple semipartial correlation coefficient is of interest.
order to estimate coverage probability. For each replication, six 95% CIs were constructed: two CIs was constructed by using two variations of (a) the asymptotic method; (b) bootstrapping and (c) bootstrapping ΔR 2 = R 2f − Rr2 ; ΔRc2 = R 2f ,c − Rr2,c , the difference in corrected
values of R 2 , where
R 2f ,c = R 2f −
p (1 − R 2f ) , n − p −1
Rr2,c = Rr2 −
k (1 − Rr2 ) . n − k −1
and
For each CI, the proportion of the replications that contained Δρ 2 = ρ 2f − ρ r2 estimated the
Method
probability of coverage. Under multivariate normality of the predictors and criterion, the asymptotic variance of R 2f is
Coverage probability was estimated for the asymptotic and two percentile bootstrap CIs. Specifically, simulation was used to estimate coverage probability for combinations of p, k, n, ρ 2f , and ρr2 , where ρ 2f is the population
ψ 2f =
squared multiple correlation coefficient for a model with p predictors (the full model), and ρ r2 is the population squared multiple correlation coefficient for a model with k predictors (the reduced model) that are a proper subset of the p predictors. In the conditions we planned to
4 ρ 2f (1 − ρ 2f ) 2 n
(1)
(Stuart, Ord, & Arnold, 1999). The asymptotic variance of Rr2 is obtained by substituting r for f in the subscripting. According to Alf and Graf
3
C. I. FOR SQUARED MULTIPLE SEMIPARTIAL CORRELATION B (α 2 ) + 1 th estimate in the rank
(1999), the asymptotic covariance between Rr2 and R 2f is
ψ fr =
4 ρ f ρ r [.5(2 ρ r ρ f − ρ f ρ r ) n
order,
n
.
B (α 2 )
indicates
rounding B (α 2 ) to the nearest whole number; the upper limit was determined by finding the B − B (α 2 ) th estimate in the rank order.
×
(1 − ρ 2f − ρ r2 − ρ r2 / ρ 2f ) + ρ r3 / ρ 3f ]
where
(2)
.
The asymptotic variance of ΔR 2 is
In all conditions B = 1000 ; thus, the bootstrap lower limit was the 26th value and the upper limit was the 975th value in the traditional percentile bootstrap. A problem with applying the traditional percentile bootstrap to ΔR 2 will occur when Δρ 2 = 0 , that is when ρ 2f = ρ r2 . Because R 2f
σ ∞2 = ψ 2f + ψ r2 − 2ψ fr , and an asymptotically correct 100(1 − α )% CI for Δρ 2 is
ΔR 2 ± zα / 2σ ∞ ,
will infrequently equal Rr2 , the CI will almost never contain 0 and the estimated coverage probability will be near zero. To address this problem, we modified the traditional percentile bootstrap by incorporating the F test of H 0 : Δρ 2 = 0 . When the F test was not significant, the lower limit of the CI was set equal to zero; otherwise, the lower limit was determined by using the traditional percentile bootstrap. To apply the traditional percentile bootstrap to ΔRc2 , steps 1 to 3 were applied with
where zα / 2 is a z critical value (Alf & Graf, 1999). In practice, the asymptotic variance is estimated by substituting R 2f and Rr2 for ρ 2f and ρ r2 , respectively. Initial results indicated that in some conditions in which ρ 2f = ρ r2 , the asymptotic CI resulted in coverage probabilities above .99. To address this problem, the lower limit of the asymptotic CI was modified. Specifically, if the lower limit of the traditional asymptotic CI was less than or equal to zero, but the F test of H 0 : Δρ 2 = 0 was significant, the lower limit was set to a small value larger than zero. In our simulations the lower limit was set equal to .001. To apply the traditional percentile bootstrap, as described in Wilcox (2003), to ΔR 2 the following steps were completed, with the first two steps completed B times.
ΔRc2 replacing ΔR 2 . Initial results indicated that in some conditions in which ρ 2f = ρ r2 , this procedure resulted in coverage probabilities above .99. To address this problem, the lower limit of the bootstrap CI was modified. Specifically, if the lower limit of the traditional CI was less than or equal to zero, but the F test of H 0 : Δρ 2 = 0 was significant, the lower limit was set to a small value larger than zero. In our simulations the lower limit was set equal to .001. A Visual Basic 6.0 program that computes the traditional and modified percentile bootstrap CIs Δρ 2 for is available at http://plaza.ufl.edu/algina/index.programs.html The multiple regression model is Y = α + β1 X 1 + β 2 X 2 + + β p X p + ε . There is
1. A sample of size n was randomly selected with replacement from the simulated participants. 2. ΔR 2 was calculated for the sample drawn in step1. 3. Once the B values of ΔR 2 were obtained, they were ranked from low to high. The lower limit of the 100 (1 − α ) % CI for Δρ 2 was determined by finding the
no loss in generality if α = 0 and/or if the variances of the dependent variable and of the independent variables are set equal to 1.0.
4
ALGINA, KESELMAN, & PENFIELD According to Browne (1969, 1975), given any set of predictors that has a squared multiple correlation coefficient of ρ 2 with Y, it is always possible to transform the predictors so that (a) the independent variables are mutually uncorrelated and (b) the regression coefficients are equal to any set of values such that p
β
2 j
Results The traditional bootstrap CI using ΔR 2 was modified by setting the lower limit to zero if the F test of H 0 : Δρ 2 = 0 was not significant; otherwise, the lower limit was determined by using the traditional percentile bootstrap. Although the modification was designed to improve performance when Δρ 2 was zero, the modification could affect coverage probability of the modified CI when Δρ 2 was zero or larger. Thus, it was important to determine if coverage probability of the bootstrap CI using ΔR 2 was affected by the modification when Δρ 2 was not equal to zero. To do this, we
= σ y2 ρ 2 .
j =1
Therefore, in the simulations (a) all variables had variance equal to one, (b) the independent variables were mutually uncorrelated, and (c) β1 = = β k −1 = 0, β k = ρ r , β k +1 = = β p −1 = 0, and β p = ρ 2f − ρ r2 . Thus, the squared multiple correlation coefficient was ρ r2 for variables X 1 to X k and
focused on the conditions in which Δρ 2 > 0 . When Δρ 2 > 0, for each combination of p,
ρ 2f for variables X 1 to X p . The data were simulated by using the following steps. 1. Generate an n × p matrix of random variables. Each of the p variables was normally distributed with mean zero and standard deviation one. All np scores were generated to be statistically independent. This matrix is X, the matrix of scores on the independent variables. 2. Generate an n × 1 vector of normally distributed random variables with mean zero and standard deviation one. All n scores were generated to be statistically independent and to be independent of the scores in X. Multiply the generated vector by
p − k , and n, there are 72 combinations of ρ r2
and ρ 2f . For each combination we tabulated the number of times that the estimated coverage probability was in the interval [.925, .975] for the traditional and modified versions of the bootstrap CI using ΔR 2 . Results indicated that the modification did not reduce the number of conditions in which the interval contained the estimated coverage probability. The asymptotic CI and traditional bootstrap CI using ΔRc2 were modified by changing the lower limit to .001 when the traditional lower limit was less or equal to zero and the F test of H 0 : Δρ 2 = 0 was significant. For our values of ρ r2 and ρ 2f , this modification could only affect the performance of these CIs when Δρ 2 was zero (and in general
1 − ρ 2f . The resulting vector
is ε , the vector of residuals. 3. Construct the p × 1 vector β in which elements 1 to k − 1 are zero and the next element is ρ r , the elements k + 1 to p − 1 are zero, and the last element is
could only affect the CI if
Δρ 2 < .001) .
Therefore, the following results describe performance of the modified CIs. For each combination of p, p − k , and n, we tabulated the number of times out of 78 possible combinations of ρ r2 and ρ 2f in which
ρ 2f − ρ r2 .
the estimated coverage probability was in the interval [.925, .975] for the modified version of the CIs. The results are presented in Table 1. For each combination, the bold value indicates the
4. Calculate n × 1 scores on the dependent variable by using y = Xβ + ε .
5
C. I. FOR SQUARED MULTIPLE SEMIPARTIAL CORRELATION method(s) that best controlled probability coverage. Ties between methods are indicated by an underlined bolded result. For p − k = 2, there were combinations of p and n in which the bootstrap CI using ΔR 2 performed as well or better than the other CIs. However, for other values of p − k , either the asymptotic or
was the asymptotic CI (See Table 3) for Δρ 2 ≥ .03 and p − k = 2. In regard to sample size required to achieve good control of coverage probability, the following comments apply to all combinations of p and p − k , with the exception of conditions in which there were nine predictors in the full model and no more than two in the reduced model. When Δρ 2 ≤ .02, the sample size required for at least one of the methods to be effective was 200 in some conditions. When Δρ 2 ≥ .03, a sample size of 50 to 100 was sufficient for at least one of the methods to be effective.
bootstrap CI using ΔRc2 performed better than the bootstrap CI using ΔR 2 . The relative performance of the modified asymptotic bootstrap and the modified percentile bootstrap using ΔRc2 depended on p − k . When p − k = 3, the modified percentile bootstrap tended to work better if p ≤ 7 ; otherwise, the two procedures worked about equally well. The modified percentile bootstrap tended to work better when p − k was between 4 or 5. When p − k = 6, the two procedures worked about equally well, particularly when the sample size was at least 150. For p − k larger than six, a value that could only occur in our design with 8 or 9 predictors in the full model, the modified asymptotic bootstrap had better control of coverage probability. Inspection of the results suggests that for p − k = 2, the relative performance of the modified asymptotic CI and the modified bootstrap CI using ΔR 2 depends on Δρ 2 . The results also suggest that larger sample sizes are required to achieve control of coverage probability when Δρ 2 is small. To illustrate these effects, we tabulated (Table 2) the number of times out of the 18 possible combinations of ρr2 and ρ 2f for Δρ 2 ≤ .02 and the number of
Discussion We investigated coverage probability for the asymptotic CI and two percentile bootstrap CIs for Δρ 2 in multiple linear regression analyses when predictors and criterion were normally distributed and Δρ 2 described the strength of association for several predictors. We also investigated modified versions of these CIs. In general, the modified methods worked at least as well as their unmodified counterparts. Specifically, results indicated that the traditional and modified bootstrap CI using ΔR 2 performed poorly, except when p − k = 2. Algina, et al. (2007) reported that when Δρ 2 describes the strength of association for one predictor ( p − k = 1) , using the modified percentile bootstrap with ΔR 2 to set a CI for Δρ 2 resulted in good coverage probability in a wide range of conditions. Thus, the results, for the case in which p − k = 1 , do not generalize to
times out of the 60 possible combinations of ρ r2
the cases in which Δρ 2 describes the strength of association for more than one predictor ( p − k > 2). The traditional and modified asymptotic CIs worked well in a variety of conditions. These results are contrary to results reported by Algina and Moulder (2001) who investigated the case of p − k = 1 and reported that the traditional asymptotic CI tended to work poorly in many conditions with sample sizes of 200 or
and ρ 2f for Δρ 2 ≥ .03 (Table 3), that the estimated coverage probability was in the interval [.925, .975]. When Δρ 2 ≤ .02 and p − k = 2, neither the modified bootstrap CI using ΔR 2 nor the modified asymptotic CI was consistently more effective than the other over all values of p. Temporarily defining effective as all estimated coverage probabilities within the interval [.925, .975] , the modified bootstrap CI using ΔR 2 was effective at a smaller size than
6
ALGINA, KESELMAN, & PENFIELD
Table 1. Number of Coverage Probability Estimates (out of 78) Inside the [.925, .975] Interval for the Modified CIs Modified Modified Modified Bootstrap on ΔR 2
Asymptotic
Bootstrap on ΔRc2
n
n
n
p
p−k
50
100
150
200
50
100
150
200
50
100
150
200
3
2
55
71
76
75
43
71
76
78
59
61
69
69
5
2
66
73
76
77
26
61
74
78
47
63
67
69
3
20
36
48
55
66
72
74
77
78
78
77
78
4
9
11
13
22
70
71
72
71
78
78
78
78
2
68
73
76
78
23
35
70
78
33
60
67
68
3
37
46
56
60
64
72
74
78
77
76
76
77
4
13
16
25
28
73
71
72
72
78
78
78
78
5
9
8
10
12
54
75
77
76
77
78
78
78
6
10
6
8
9
29
73
78
78
18
70
78
78
2
70
75
77
78
23
24
67
74
30
59
65
69
3
41
51
57
60
60
72
75
77
67
74
74
76
4
20
21
28
34
71
73
72
72
78
78
78
78
5
13
10
13
15
53
78
76
76
77
78
78
78
6
10
6
7
11
25
72
77
78
23
73
78
7
10
6
6
7
12
42
76
78
4
26
54
64
2
62
75
77
78
17
20
57
74
28
58
66
69
3
45
52
58
62
63
69
75
78
56
73
72
77
4
23
25
29
38
71
72
72
72
78
78
78
78
5
13
12
15
19
51
77
77
77
77
78
78
78
6
10
8
9
11
28
74
78
78
37
73
78
78
7
12
6
7
9
24
45
66
69
23
41
55
59
8
10
6
6
6
5
19
38
55
1
7
27
38
7
8
9
7
78
C. I. FOR SQUARED MULTIPLE SEMIPARTIAL CORRELATION
Table 2. Number of Coverage Probability Estimates (out of 18) Inside the [.925, .975] Interval for Modified CIs and Δρ 2 ≤ .02 Modified
Modified
Modified
Bootstrap on ΔR 2
Asymptotic
Bootstrap on ΔRc2
n
n
n
p
p−k
50
100
150
200
50
100
150
200
50
100
150
200
3
2
6
11
16
15
12
14
17
18
7
7
10
9
2
8
13
16
17
12
16
17
18
6
7
8
9
3
6
6
6
6
9
12
14
17
18
18
17
18
4
8
6
6
6
15
11
12
11
18
18
18
18
2
8
13
16
18
12
14
16
18
6
6
8
8
3
8
6
6
7
13
12
14
18
18
16
16
17
4
7
6
6
6
17
12
12
12
18
18
18
18
5
8
6
6
6
15
17
17
16
17
18
18
18
6
10
6
6
6
5
18
18
18
0
11
18
18
2
11
15
17
18
12
15
17
17
6
6
7
9
3
7
6
6
6
13
12
15
17
17
15
14
16
4
9
6
6
6
17
13
12
12
18
18
18
18
5
10
6
6
6
13
18
16
16
17
18
18
18
6
10
6
6
6
4
18
18
18
0
13
18
18
7
10
6
6
6
0
10
18
18
0
0
1
4
2
15
15
17
18
12
15
15
18
6
6
8
9
3
7
6
6
7
13
13
15
18
16
13
12
17
4
8
6
6
6
17
12
12
12
18
18
18
18
5
9
6
6
6
13
17
17
17
17
18
18
18
6
9
6
6
6
4
18
18
18
0
13
18
18
7
12
6
6
6
0
8
17
18
0
0
2
6
8
10
6
6
6
0
1
4
8
0
0
0
0
5
7
8
9
8
ALGINA, KESELMAN, & PENFIELD
Table 3. Number of Coverage Probability Estimates (out of 60) Inside the [.925, .975] Interval for Modified CIs and Δρ 2 ≥ .03 Modified
Modified
Modified
Bootstrap on ΔR 2
Asymptotic
Bootstrap on ΔRc2
n
n
n
p
p−k
50
100
150
200
50
100
150
200
50
100
150
200
3
2
49
60
60
60
31
57
59
60
52
54
59
60
2
58
60
60
60
14
45
57
60
41
56
59
60
3
14
30
42
49
57
60
60
60
60
60
60
60
4
1
5
7
16
55
60
60
60
60
60
60
60
2
60
60
60
60
11
21
54
60
27
54
59
60
3
29
40
50
53
51
60
60
60
59
60
60
60
4
6
10
19
22
56
59
60
60
60
60
60
60
5
1
2
4
6
39
58
60
60
60
60
60
60
6
0
0
2
3
24
55
60
60
18
59
60
60
2
59
60
60
60
11
9
50
57
24
53
58
60
3
34
45
51
54
47
60
60
60
50
59
60
60
4
11
15
22
28
54
60
60
60
60
60
60
60
5
3
4
7
9
40
60
60
60
60
60
60
60
6
0
0
1
5
21
54
59
60
23
60
60
60
0
0
0
1
12
32
58
60
4
26
53
60
2
47
60
60
5
5
42
56
22
52
58
60
3
38
46
52
55
50
56
60
60
40
60
60
60
4
15
19
23
32
54
60
60
60
60
60
60
60
5
4
6
9
13
38
60
60
60
60
60
60
60
6
1
2
3
5
24
56
60
60
37
60
60
60
7
0
0
1
3
24
37
49
51
23
41
53
53
8
0
0
0
0
5
18
34
47
1
7
27
38
5
7
8
7 9
60
9
C. I. FOR SQUARED MULTIPLE SEMIPARTIAL CORRELATION the sample size should 200 if p = 8 and should be larger than 200 if p = 9.
less. Algina and Moulder did not report results on the modified asymptotic CI, but the modification used in this paper is designed to improve performance when Δρ 2 is zero, so that it is unlikely that using the modification with p − k = 1 would overcome the problems that Algina and Moulder reported. Unfortunately, although the modified asymptotic CI and the modified bootstrap CI on ΔRc2 worked better than the competitors investigated in this study, neither of these CIs worked well in all of the conditions we investigated. Particularly problematic was the condition in which the number of predictors in the full model was nine and the number of predictors in the reduced model was no larger than two. Defining adequate control of coverage probability by at least 77 of the 78 combinations of ρ r2 and ρ 2f for each combination of p and
References Alf, E. F., Jr., & Graf, R. G. (1999). Asymptotic confidence limits for the difference between two squared multiple correlations: A simplified approach. Psychological Methods, 4, 70-75. Algina, J., Keselman, H. J., & Penfield, R. D. (2007). Confidence intervals for an effect size measure in multiple linear regression. Educational and Psychological Measurement, 67, 207-218. Algina, J., & Moulder, B. C. (2001). Sample sizes for confidence intervals on the increase in the squared multiple correlation coefficient. Educational and Psychological Measurement, 61, 633-649. Browne, M. W. (1969). Precision of prediction (Research Bulletin 69-69). Princeton, NJ: Educational Testing Service. Browne, M. W. (1975). Predictive validity of a linear regression equation. British Journal of Mathematical & Statistical Psychology, 28, 79-87. Cohen, J, Cohen, P, West, S. G., & Aiken, L. S. (2003). Applied multiple regression and correlation for the behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum. Graf, R. G., & Alf, E. F., Jr. (1999). Asymptotic confidence limits for partial and squared partial correlations. Applied Psychological Measurement, 23, 116-119. Hedges, L. V., & Olkin, I. (1981). The asymptotic distribution of commonality components. Psychometrika, 46, 331-336. Mood, A. M. (1969). Macro-analysis of the American educational system. Operations Research, 17, 770-784. Mood, A. M. (1971). Partitioning variance in multiple regression analyses as a tool for developing learning models. American Educational Research Journal, 8, 191-202. Pedhazur, E. J. (1997). Multiple regression in behavioral research (3rd ed.). Belmont. CA: Wadsworth. Olkin, I., & Finn, J. D. (1995). Correlations redux. Psychological Bulletin, 188, 155-164. Stuart, A., Ord, J. K., & Arnold, S. (1999). The advanced theory of statistics (6th ed., Vol. 2A). New York: Oxford University Press. Wilcox, R. R. (2003). Applying contemporary statistical techniques. San Diego: Academic Press.
predictors in the reduced model was no larger than two. Defining adequate control of coverage probability by at least 77 of the 78 combinations of ρ r2 and ρ 2f for each combination of p and p − k , we offer the following recommendations for a CI method and sample size in order to achieve adequate control of coverage probability,: (a) If p − k = 2 and p ≤ 7, the modified asymptotic CI should be used with a sample size of at least 200. For p ≥ 8, a sample size of 150
and the modified bootstrap CI using ΔR 2 should be used; (b) If p − k = 3 and p ≤ 7, the modified bootstrap CI using ΔRc2 should be used with a sample size of at least 50. If p ≥ 8 the modified asymptotic CI should be used with a sample size of at least 200; (c) If p − k = 4 or 5 and p ≤ 9, the modified bootstrap CI using ΔRc2 should be used with a sample size of at least 50; (d) If p − k = 6 (and therefore p ≥ 7 ) , the modified bootstrap CI using ΔRc2 should be used with a sample size of at least 150; (e) If p − k ≥ 7 (and therefore p ≥ 8 ) , the modified asymptotic CI should be used and
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